Journal of Probability and Statistics
Volume 2009 (2009), Article ID 275308, 19 pages
Research Article

Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise

Department of Applied Probability and Statistics, University of California, Santa Barbara, Santa Barbara, CA 93106-3110, USA

Received 30 July 2009; Accepted 16 December 2009

Academic Editor: A. T. A. Wood

Copyright © 2009 Andrew V. Carter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We find asymptotically sufficient statistics that could help simplify inference in nonparametric regression problems with correlated errors. These statistics are derived from a wavelet decomposition that is used to whiten the noise process and to effectively separate high-resolution and low-resolution components. The lower-resolution components contain nearly all the available information about the mean function, and the higher-resolution components can be used to estimate the error covariances. The strength of the correlation among the errors is related to the speed at which the variance of the higher-resolution components shrinks, and this is considered an additional nuisance parameter in the model. We show that the NPR experiment with correlated noise is asymptotically equivalent to an experiment that observes the mean function in the presence of a continuous Gaussian process that is similar to a fractional Brownian motion. These results provide a theoretical motivation for some commonly proposed wavelet estimation techniques.